On a new family of complete $G_2$-holonomy Riemannian metrics on $S^3\times\mathbb R^4$

Studying a system of first-order nonlinear ordinary differential equations for the functions determining a deformation of the standard conic metric over $S^3\times S^3$, we prove the existence of a one-parameter family of complete $G_2$-holonomy Riemannian metrics on $S^3\times\mathbb R^4$.